# Properties

 Label 2640.2.a.bb.1.1 Level $2640$ Weight $2$ Character 2640.1 Self dual yes Analytic conductor $21.081$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.0805061336$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2640.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} +1.00000 q^{11} -5.65685 q^{13} -1.00000 q^{15} -1.17157 q^{17} +6.82843 q^{19} -0.828427 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.82843 q^{29} +1.00000 q^{33} +0.828427 q^{35} +11.6569 q^{37} -5.65685 q^{39} +4.82843 q^{41} +8.82843 q^{43} -1.00000 q^{45} +4.00000 q^{47} -6.31371 q^{49} -1.17157 q^{51} +9.31371 q^{53} -1.00000 q^{55} +6.82843 q^{57} +4.00000 q^{59} -11.6569 q^{61} -0.828427 q^{63} +5.65685 q^{65} +5.65685 q^{67} +4.00000 q^{69} -2.34315 q^{71} +11.3137 q^{73} +1.00000 q^{75} -0.828427 q^{77} -8.48528 q^{79} +1.00000 q^{81} +10.0000 q^{83} +1.17157 q^{85} -4.82843 q^{87} +3.65685 q^{89} +4.68629 q^{91} -6.82843 q^{95} +11.6569 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{5} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{5} + 4q^{7} + 2q^{9} + 2q^{11} - 2q^{15} - 8q^{17} + 8q^{19} + 4q^{21} + 8q^{23} + 2q^{25} + 2q^{27} - 4q^{29} + 2q^{33} - 4q^{35} + 12q^{37} + 4q^{41} + 12q^{43} - 2q^{45} + 8q^{47} + 10q^{49} - 8q^{51} - 4q^{53} - 2q^{55} + 8q^{57} + 8q^{59} - 12q^{61} + 4q^{63} + 8q^{69} - 16q^{71} + 2q^{75} + 4q^{77} + 2q^{81} + 20q^{83} + 8q^{85} - 4q^{87} - 4q^{89} + 32q^{91} - 8q^{95} + 12q^{97} + 2q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −0.828427 −0.313116 −0.156558 0.987669i $$-0.550040\pi$$
−0.156558 + 0.987669i $$0.550040\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −5.65685 −1.56893 −0.784465 0.620174i $$-0.787062\pi$$
−0.784465 + 0.620174i $$0.787062\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −1.17157 −0.284148 −0.142074 0.989856i $$-0.545377\pi$$
−0.142074 + 0.989856i $$0.545377\pi$$
$$18$$ 0 0
$$19$$ 6.82843 1.56655 0.783274 0.621676i $$-0.213548\pi$$
0.783274 + 0.621676i $$0.213548\pi$$
$$20$$ 0 0
$$21$$ −0.828427 −0.180778
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −4.82843 −0.896616 −0.448308 0.893879i $$-0.647973\pi$$
−0.448308 + 0.893879i $$0.647973\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 1.00000 0.174078
$$34$$ 0 0
$$35$$ 0.828427 0.140030
$$36$$ 0 0
$$37$$ 11.6569 1.91638 0.958188 0.286141i $$-0.0923726\pi$$
0.958188 + 0.286141i $$0.0923726\pi$$
$$38$$ 0 0
$$39$$ −5.65685 −0.905822
$$40$$ 0 0
$$41$$ 4.82843 0.754074 0.377037 0.926198i $$-0.376943\pi$$
0.377037 + 0.926198i $$0.376943\pi$$
$$42$$ 0 0
$$43$$ 8.82843 1.34632 0.673161 0.739496i $$-0.264936\pi$$
0.673161 + 0.739496i $$0.264936\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ −6.31371 −0.901958
$$50$$ 0 0
$$51$$ −1.17157 −0.164053
$$52$$ 0 0
$$53$$ 9.31371 1.27934 0.639668 0.768651i $$-0.279072\pi$$
0.639668 + 0.768651i $$0.279072\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 6.82843 0.904447
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −11.6569 −1.49251 −0.746254 0.665662i $$-0.768149\pi$$
−0.746254 + 0.665662i $$0.768149\pi$$
$$62$$ 0 0
$$63$$ −0.828427 −0.104372
$$64$$ 0 0
$$65$$ 5.65685 0.701646
$$66$$ 0 0
$$67$$ 5.65685 0.691095 0.345547 0.938401i $$-0.387693\pi$$
0.345547 + 0.938401i $$0.387693\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −2.34315 −0.278080 −0.139040 0.990287i $$-0.544402\pi$$
−0.139040 + 0.990287i $$0.544402\pi$$
$$72$$ 0 0
$$73$$ 11.3137 1.32417 0.662085 0.749429i $$-0.269672\pi$$
0.662085 + 0.749429i $$0.269672\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −0.828427 −0.0944080
$$78$$ 0 0
$$79$$ −8.48528 −0.954669 −0.477334 0.878722i $$-0.658397\pi$$
−0.477334 + 0.878722i $$0.658397\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 10.0000 1.09764 0.548821 0.835940i $$-0.315077\pi$$
0.548821 + 0.835940i $$0.315077\pi$$
$$84$$ 0 0
$$85$$ 1.17157 0.127075
$$86$$ 0 0
$$87$$ −4.82843 −0.517662
$$88$$ 0 0
$$89$$ 3.65685 0.387626 0.193813 0.981039i $$-0.437915\pi$$
0.193813 + 0.981039i $$0.437915\pi$$
$$90$$ 0 0
$$91$$ 4.68629 0.491257
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6.82843 −0.700582
$$96$$ 0 0
$$97$$ 11.6569 1.18357 0.591787 0.806094i $$-0.298423\pi$$
0.591787 + 0.806094i $$0.298423\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −0.828427 −0.0824316 −0.0412158 0.999150i $$-0.513123\pi$$
−0.0412158 + 0.999150i $$0.513123\pi$$
$$102$$ 0 0
$$103$$ 3.31371 0.326509 0.163255 0.986584i $$-0.447801\pi$$
0.163255 + 0.986584i $$0.447801\pi$$
$$104$$ 0 0
$$105$$ 0.828427 0.0808462
$$106$$ 0 0
$$107$$ −17.3137 −1.67378 −0.836890 0.547372i $$-0.815628\pi$$
−0.836890 + 0.547372i $$0.815628\pi$$
$$108$$ 0 0
$$109$$ 17.3137 1.65835 0.829176 0.558987i $$-0.188810\pi$$
0.829176 + 0.558987i $$0.188810\pi$$
$$110$$ 0 0
$$111$$ 11.6569 1.10642
$$112$$ 0 0
$$113$$ −18.9706 −1.78460 −0.892300 0.451442i $$-0.850910\pi$$
−0.892300 + 0.451442i $$0.850910\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 0 0
$$117$$ −5.65685 −0.522976
$$118$$ 0 0
$$119$$ 0.970563 0.0889713
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 4.82843 0.435365
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 14.4853 1.28536 0.642680 0.766134i $$-0.277822\pi$$
0.642680 + 0.766134i $$0.277822\pi$$
$$128$$ 0 0
$$129$$ 8.82843 0.777300
$$130$$ 0 0
$$131$$ −3.31371 −0.289520 −0.144760 0.989467i $$-0.546241\pi$$
−0.144760 + 0.989467i $$0.546241\pi$$
$$132$$ 0 0
$$133$$ −5.65685 −0.490511
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −13.3137 −1.13747 −0.568733 0.822522i $$-0.692566\pi$$
−0.568733 + 0.822522i $$0.692566\pi$$
$$138$$ 0 0
$$139$$ −0.485281 −0.0411610 −0.0205805 0.999788i $$-0.506551\pi$$
−0.0205805 + 0.999788i $$0.506551\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 0 0
$$143$$ −5.65685 −0.473050
$$144$$ 0 0
$$145$$ 4.82843 0.400979
$$146$$ 0 0
$$147$$ −6.31371 −0.520746
$$148$$ 0 0
$$149$$ 1.51472 0.124091 0.0620453 0.998073i $$-0.480238\pi$$
0.0620453 + 0.998073i $$0.480238\pi$$
$$150$$ 0 0
$$151$$ −16.4853 −1.34155 −0.670777 0.741659i $$-0.734039\pi$$
−0.670777 + 0.741659i $$0.734039\pi$$
$$152$$ 0 0
$$153$$ −1.17157 −0.0947161
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 0 0
$$159$$ 9.31371 0.738625
$$160$$ 0 0
$$161$$ −3.31371 −0.261157
$$162$$ 0 0
$$163$$ 7.31371 0.572854 0.286427 0.958102i $$-0.407532\pi$$
0.286427 + 0.958102i $$0.407532\pi$$
$$164$$ 0 0
$$165$$ −1.00000 −0.0778499
$$166$$ 0 0
$$167$$ 13.3137 1.03025 0.515123 0.857116i $$-0.327746\pi$$
0.515123 + 0.857116i $$0.327746\pi$$
$$168$$ 0 0
$$169$$ 19.0000 1.46154
$$170$$ 0 0
$$171$$ 6.82843 0.522183
$$172$$ 0 0
$$173$$ 2.82843 0.215041 0.107521 0.994203i $$-0.465709\pi$$
0.107521 + 0.994203i $$0.465709\pi$$
$$174$$ 0 0
$$175$$ −0.828427 −0.0626232
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ 17.6569 1.31974 0.659868 0.751382i $$-0.270612\pi$$
0.659868 + 0.751382i $$0.270612\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ −11.6569 −0.861699
$$184$$ 0 0
$$185$$ −11.6569 −0.857029
$$186$$ 0 0
$$187$$ −1.17157 −0.0856739
$$188$$ 0 0
$$189$$ −0.828427 −0.0602592
$$190$$ 0 0
$$191$$ −5.65685 −0.409316 −0.204658 0.978834i $$-0.565608\pi$$
−0.204658 + 0.978834i $$0.565608\pi$$
$$192$$ 0 0
$$193$$ 13.6569 0.983042 0.491521 0.870866i $$-0.336441\pi$$
0.491521 + 0.870866i $$0.336441\pi$$
$$194$$ 0 0
$$195$$ 5.65685 0.405096
$$196$$ 0 0
$$197$$ −8.48528 −0.604551 −0.302276 0.953221i $$-0.597746\pi$$
−0.302276 + 0.953221i $$0.597746\pi$$
$$198$$ 0 0
$$199$$ 21.6569 1.53521 0.767607 0.640921i $$-0.221447\pi$$
0.767607 + 0.640921i $$0.221447\pi$$
$$200$$ 0 0
$$201$$ 5.65685 0.399004
$$202$$ 0 0
$$203$$ 4.00000 0.280745
$$204$$ 0 0
$$205$$ −4.82843 −0.337232
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 6.82843 0.472332
$$210$$ 0 0
$$211$$ −1.17157 −0.0806544 −0.0403272 0.999187i $$-0.512840\pi$$
−0.0403272 + 0.999187i $$0.512840\pi$$
$$212$$ 0 0
$$213$$ −2.34315 −0.160550
$$214$$ 0 0
$$215$$ −8.82843 −0.602094
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 11.3137 0.764510
$$220$$ 0 0
$$221$$ 6.62742 0.445808
$$222$$ 0 0
$$223$$ 6.34315 0.424768 0.212384 0.977186i $$-0.431877\pi$$
0.212384 + 0.977186i $$0.431877\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 14.0000 0.929213 0.464606 0.885517i $$-0.346196\pi$$
0.464606 + 0.885517i $$0.346196\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ −0.828427 −0.0545065
$$232$$ 0 0
$$233$$ −18.8284 −1.23349 −0.616746 0.787163i $$-0.711549\pi$$
−0.616746 + 0.787163i $$0.711549\pi$$
$$234$$ 0 0
$$235$$ −4.00000 −0.260931
$$236$$ 0 0
$$237$$ −8.48528 −0.551178
$$238$$ 0 0
$$239$$ −17.6569 −1.14213 −0.571063 0.820906i $$-0.693469\pi$$
−0.571063 + 0.820906i $$0.693469\pi$$
$$240$$ 0 0
$$241$$ −12.3431 −0.795092 −0.397546 0.917582i $$-0.630138\pi$$
−0.397546 + 0.917582i $$0.630138\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 6.31371 0.403368
$$246$$ 0 0
$$247$$ −38.6274 −2.45780
$$248$$ 0 0
$$249$$ 10.0000 0.633724
$$250$$ 0 0
$$251$$ −20.9706 −1.32365 −0.661825 0.749658i $$-0.730218\pi$$
−0.661825 + 0.749658i $$0.730218\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ 0 0
$$255$$ 1.17157 0.0733667
$$256$$ 0 0
$$257$$ −16.3431 −1.01946 −0.509729 0.860335i $$-0.670254\pi$$
−0.509729 + 0.860335i $$0.670254\pi$$
$$258$$ 0 0
$$259$$ −9.65685 −0.600048
$$260$$ 0 0
$$261$$ −4.82843 −0.298872
$$262$$ 0 0
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ −9.31371 −0.572137
$$266$$ 0 0
$$267$$ 3.65685 0.223796
$$268$$ 0 0
$$269$$ 20.6274 1.25768 0.628838 0.777536i $$-0.283531\pi$$
0.628838 + 0.777536i $$0.283531\pi$$
$$270$$ 0 0
$$271$$ 11.7990 0.716738 0.358369 0.933580i $$-0.383333\pi$$
0.358369 + 0.933580i $$0.383333\pi$$
$$272$$ 0 0
$$273$$ 4.68629 0.283627
$$274$$ 0 0
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −2.34315 −0.140786 −0.0703930 0.997519i $$-0.522425\pi$$
−0.0703930 + 0.997519i $$0.522425\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −11.1716 −0.666440 −0.333220 0.942849i $$-0.608135\pi$$
−0.333220 + 0.942849i $$0.608135\pi$$
$$282$$ 0 0
$$283$$ 8.82843 0.524796 0.262398 0.964960i $$-0.415487\pi$$
0.262398 + 0.964960i $$0.415487\pi$$
$$284$$ 0 0
$$285$$ −6.82843 −0.404481
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ 0 0
$$289$$ −15.6274 −0.919260
$$290$$ 0 0
$$291$$ 11.6569 0.683337
$$292$$ 0 0
$$293$$ −6.82843 −0.398921 −0.199460 0.979906i $$-0.563919\pi$$
−0.199460 + 0.979906i $$0.563919\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 0 0
$$299$$ −22.6274 −1.30858
$$300$$ 0 0
$$301$$ −7.31371 −0.421555
$$302$$ 0 0
$$303$$ −0.828427 −0.0475919
$$304$$ 0 0
$$305$$ 11.6569 0.667470
$$306$$ 0 0
$$307$$ 3.17157 0.181011 0.0905056 0.995896i $$-0.471152\pi$$
0.0905056 + 0.995896i $$0.471152\pi$$
$$308$$ 0 0
$$309$$ 3.31371 0.188510
$$310$$ 0 0
$$311$$ 3.31371 0.187903 0.0939516 0.995577i $$-0.470050\pi$$
0.0939516 + 0.995577i $$0.470050\pi$$
$$312$$ 0 0
$$313$$ 15.6569 0.884978 0.442489 0.896774i $$-0.354096\pi$$
0.442489 + 0.896774i $$0.354096\pi$$
$$314$$ 0 0
$$315$$ 0.828427 0.0466766
$$316$$ 0 0
$$317$$ −26.2843 −1.47627 −0.738136 0.674652i $$-0.764294\pi$$
−0.738136 + 0.674652i $$0.764294\pi$$
$$318$$ 0 0
$$319$$ −4.82843 −0.270340
$$320$$ 0 0
$$321$$ −17.3137 −0.966357
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ −5.65685 −0.313786
$$326$$ 0 0
$$327$$ 17.3137 0.957450
$$328$$ 0 0
$$329$$ −3.31371 −0.182691
$$330$$ 0 0
$$331$$ −6.34315 −0.348651 −0.174325 0.984688i $$-0.555774\pi$$
−0.174325 + 0.984688i $$0.555774\pi$$
$$332$$ 0 0
$$333$$ 11.6569 0.638792
$$334$$ 0 0
$$335$$ −5.65685 −0.309067
$$336$$ 0 0
$$337$$ 3.31371 0.180509 0.0902546 0.995919i $$-0.471232\pi$$
0.0902546 + 0.995919i $$0.471232\pi$$
$$338$$ 0 0
$$339$$ −18.9706 −1.03034
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 11.0294 0.595534
$$344$$ 0 0
$$345$$ −4.00000 −0.215353
$$346$$ 0 0
$$347$$ 29.3137 1.57364 0.786821 0.617181i $$-0.211725\pi$$
0.786821 + 0.617181i $$0.211725\pi$$
$$348$$ 0 0
$$349$$ 10.9706 0.587241 0.293620 0.955922i $$-0.405140\pi$$
0.293620 + 0.955922i $$0.405140\pi$$
$$350$$ 0 0
$$351$$ −5.65685 −0.301941
$$352$$ 0 0
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 0 0
$$355$$ 2.34315 0.124361
$$356$$ 0 0
$$357$$ 0.970563 0.0513676
$$358$$ 0 0
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 27.6274 1.45407
$$362$$ 0 0
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ −11.3137 −0.592187
$$366$$ 0 0
$$367$$ −9.65685 −0.504084 −0.252042 0.967716i $$-0.581102\pi$$
−0.252042 + 0.967716i $$0.581102\pi$$
$$368$$ 0 0
$$369$$ 4.82843 0.251358
$$370$$ 0 0
$$371$$ −7.71573 −0.400581
$$372$$ 0 0
$$373$$ 10.6274 0.550267 0.275133 0.961406i $$-0.411278\pi$$
0.275133 + 0.961406i $$0.411278\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 27.3137 1.40673
$$378$$ 0 0
$$379$$ 23.3137 1.19754 0.598772 0.800919i $$-0.295655\pi$$
0.598772 + 0.800919i $$0.295655\pi$$
$$380$$ 0 0
$$381$$ 14.4853 0.742103
$$382$$ 0 0
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 0 0
$$385$$ 0.828427 0.0422206
$$386$$ 0 0
$$387$$ 8.82843 0.448774
$$388$$ 0 0
$$389$$ −23.6569 −1.19945 −0.599725 0.800206i $$-0.704723\pi$$
−0.599725 + 0.800206i $$0.704723\pi$$
$$390$$ 0 0
$$391$$ −4.68629 −0.236996
$$392$$ 0 0
$$393$$ −3.31371 −0.167154
$$394$$ 0 0
$$395$$ 8.48528 0.426941
$$396$$ 0 0
$$397$$ −14.9706 −0.751351 −0.375676 0.926751i $$-0.622589\pi$$
−0.375676 + 0.926751i $$0.622589\pi$$
$$398$$ 0 0
$$399$$ −5.65685 −0.283197
$$400$$ 0 0
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 11.6569 0.577809
$$408$$ 0 0
$$409$$ −19.6569 −0.971969 −0.485984 0.873967i $$-0.661539\pi$$
−0.485984 + 0.873967i $$0.661539\pi$$
$$410$$ 0 0
$$411$$ −13.3137 −0.656717
$$412$$ 0 0
$$413$$ −3.31371 −0.163057
$$414$$ 0 0
$$415$$ −10.0000 −0.490881
$$416$$ 0 0
$$417$$ −0.485281 −0.0237643
$$418$$ 0 0
$$419$$ 36.9706 1.80613 0.903065 0.429504i $$-0.141311\pi$$
0.903065 + 0.429504i $$0.141311\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 0 0
$$423$$ 4.00000 0.194487
$$424$$ 0 0
$$425$$ −1.17157 −0.0568296
$$426$$ 0 0
$$427$$ 9.65685 0.467328
$$428$$ 0 0
$$429$$ −5.65685 −0.273115
$$430$$ 0 0
$$431$$ −21.6569 −1.04317 −0.521587 0.853198i $$-0.674660\pi$$
−0.521587 + 0.853198i $$0.674660\pi$$
$$432$$ 0 0
$$433$$ −15.6569 −0.752420 −0.376210 0.926534i $$-0.622773\pi$$
−0.376210 + 0.926534i $$0.622773\pi$$
$$434$$ 0 0
$$435$$ 4.82843 0.231505
$$436$$ 0 0
$$437$$ 27.3137 1.30659
$$438$$ 0 0
$$439$$ 20.4853 0.977709 0.488855 0.872365i $$-0.337415\pi$$
0.488855 + 0.872365i $$0.337415\pi$$
$$440$$ 0 0
$$441$$ −6.31371 −0.300653
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ −3.65685 −0.173352
$$446$$ 0 0
$$447$$ 1.51472 0.0716437
$$448$$ 0 0
$$449$$ 30.9706 1.46159 0.730796 0.682596i $$-0.239149\pi$$
0.730796 + 0.682596i $$0.239149\pi$$
$$450$$ 0 0
$$451$$ 4.82843 0.227362
$$452$$ 0 0
$$453$$ −16.4853 −0.774546
$$454$$ 0 0
$$455$$ −4.68629 −0.219697
$$456$$ 0 0
$$457$$ −23.3137 −1.09057 −0.545285 0.838251i $$-0.683578\pi$$
−0.545285 + 0.838251i $$0.683578\pi$$
$$458$$ 0 0
$$459$$ −1.17157 −0.0546843
$$460$$ 0 0
$$461$$ 0.142136 0.00661992 0.00330996 0.999995i $$-0.498946\pi$$
0.00330996 + 0.999995i $$0.498946\pi$$
$$462$$ 0 0
$$463$$ −4.97056 −0.231002 −0.115501 0.993307i $$-0.536847\pi$$
−0.115501 + 0.993307i $$0.536847\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.6274 1.04707 0.523536 0.852004i $$-0.324613\pi$$
0.523536 + 0.852004i $$0.324613\pi$$
$$468$$ 0 0
$$469$$ −4.68629 −0.216393
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 8.82843 0.405932
$$474$$ 0 0
$$475$$ 6.82843 0.313310
$$476$$ 0 0
$$477$$ 9.31371 0.426445
$$478$$ 0 0
$$479$$ −36.9706 −1.68923 −0.844614 0.535376i $$-0.820170\pi$$
−0.844614 + 0.535376i $$0.820170\pi$$
$$480$$ 0 0
$$481$$ −65.9411 −3.00666
$$482$$ 0 0
$$483$$ −3.31371 −0.150779
$$484$$ 0 0
$$485$$ −11.6569 −0.529310
$$486$$ 0 0
$$487$$ 12.9706 0.587752 0.293876 0.955844i $$-0.405055\pi$$
0.293876 + 0.955844i $$0.405055\pi$$
$$488$$ 0 0
$$489$$ 7.31371 0.330737
$$490$$ 0 0
$$491$$ −14.3431 −0.647297 −0.323649 0.946177i $$-0.604910\pi$$
−0.323649 + 0.946177i $$0.604910\pi$$
$$492$$ 0 0
$$493$$ 5.65685 0.254772
$$494$$ 0 0
$$495$$ −1.00000 −0.0449467
$$496$$ 0 0
$$497$$ 1.94113 0.0870714
$$498$$ 0 0
$$499$$ 22.3431 1.00022 0.500108 0.865963i $$-0.333294\pi$$
0.500108 + 0.865963i $$0.333294\pi$$
$$500$$ 0 0
$$501$$ 13.3137 0.594813
$$502$$ 0 0
$$503$$ −17.3137 −0.771980 −0.385990 0.922503i $$-0.626140\pi$$
−0.385990 + 0.922503i $$0.626140\pi$$
$$504$$ 0 0
$$505$$ 0.828427 0.0368645
$$506$$ 0 0
$$507$$ 19.0000 0.843820
$$508$$ 0 0
$$509$$ 18.6863 0.828255 0.414128 0.910219i $$-0.364087\pi$$
0.414128 + 0.910219i $$0.364087\pi$$
$$510$$ 0 0
$$511$$ −9.37258 −0.414619
$$512$$ 0 0
$$513$$ 6.82843 0.301482
$$514$$ 0 0
$$515$$ −3.31371 −0.146019
$$516$$ 0 0
$$517$$ 4.00000 0.175920
$$518$$ 0 0
$$519$$ 2.82843 0.124154
$$520$$ 0 0
$$521$$ −32.6274 −1.42943 −0.714717 0.699414i $$-0.753444\pi$$
−0.714717 + 0.699414i $$0.753444\pi$$
$$522$$ 0 0
$$523$$ 9.51472 0.416050 0.208025 0.978124i $$-0.433297\pi$$
0.208025 + 0.978124i $$0.433297\pi$$
$$524$$ 0 0
$$525$$ −0.828427 −0.0361555
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ −27.3137 −1.18309
$$534$$ 0 0
$$535$$ 17.3137 0.748537
$$536$$ 0 0
$$537$$ 17.6569 0.761950
$$538$$ 0 0
$$539$$ −6.31371 −0.271951
$$540$$ 0 0
$$541$$ 17.3137 0.744374 0.372187 0.928158i $$-0.378608\pi$$
0.372187 + 0.928158i $$0.378608\pi$$
$$542$$ 0 0
$$543$$ −14.0000 −0.600798
$$544$$ 0 0
$$545$$ −17.3137 −0.741638
$$546$$ 0 0
$$547$$ 8.14214 0.348133 0.174066 0.984734i $$-0.444309\pi$$
0.174066 + 0.984734i $$0.444309\pi$$
$$548$$ 0 0
$$549$$ −11.6569 −0.497502
$$550$$ 0 0
$$551$$ −32.9706 −1.40459
$$552$$ 0 0
$$553$$ 7.02944 0.298922
$$554$$ 0 0
$$555$$ −11.6569 −0.494806
$$556$$ 0 0
$$557$$ 5.17157 0.219127 0.109563 0.993980i $$-0.465055\pi$$
0.109563 + 0.993980i $$0.465055\pi$$
$$558$$ 0 0
$$559$$ −49.9411 −2.11228
$$560$$ 0 0
$$561$$ −1.17157 −0.0494638
$$562$$ 0 0
$$563$$ −31.6569 −1.33418 −0.667089 0.744978i $$-0.732460\pi$$
−0.667089 + 0.744978i $$0.732460\pi$$
$$564$$ 0 0
$$565$$ 18.9706 0.798098
$$566$$ 0 0
$$567$$ −0.828427 −0.0347907
$$568$$ 0 0
$$569$$ −35.4558 −1.48639 −0.743193 0.669077i $$-0.766690\pi$$
−0.743193 + 0.669077i $$0.766690\pi$$
$$570$$ 0 0
$$571$$ 16.4853 0.689888 0.344944 0.938623i $$-0.387898\pi$$
0.344944 + 0.938623i $$0.387898\pi$$
$$572$$ 0 0
$$573$$ −5.65685 −0.236318
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 0 0
$$579$$ 13.6569 0.567559
$$580$$ 0 0
$$581$$ −8.28427 −0.343689
$$582$$ 0 0
$$583$$ 9.31371 0.385734
$$584$$ 0 0
$$585$$ 5.65685 0.233882
$$586$$ 0 0
$$587$$ −14.6274 −0.603738 −0.301869 0.953349i $$-0.597611\pi$$
−0.301869 + 0.953349i $$0.597611\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −8.48528 −0.349038
$$592$$ 0 0
$$593$$ −22.8284 −0.937451 −0.468726 0.883344i $$-0.655287\pi$$
−0.468726 + 0.883344i $$0.655287\pi$$
$$594$$ 0 0
$$595$$ −0.970563 −0.0397892
$$596$$ 0 0
$$597$$ 21.6569 0.886356
$$598$$ 0 0
$$599$$ −27.3137 −1.11601 −0.558004 0.829838i $$-0.688433\pi$$
−0.558004 + 0.829838i $$0.688433\pi$$
$$600$$ 0 0
$$601$$ −5.31371 −0.216751 −0.108375 0.994110i $$-0.534565\pi$$
−0.108375 + 0.994110i $$0.534565\pi$$
$$602$$ 0 0
$$603$$ 5.65685 0.230365
$$604$$ 0 0
$$605$$ −1.00000 −0.0406558
$$606$$ 0 0
$$607$$ −1.51472 −0.0614805 −0.0307403 0.999527i $$-0.509786\pi$$
−0.0307403 + 0.999527i $$0.509786\pi$$
$$608$$ 0 0
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ −22.6274 −0.915407
$$612$$ 0 0
$$613$$ −45.9411 −1.85554 −0.927772 0.373147i $$-0.878279\pi$$
−0.927772 + 0.373147i $$0.878279\pi$$
$$614$$ 0 0
$$615$$ −4.82843 −0.194701
$$616$$ 0 0
$$617$$ 0.343146 0.0138145 0.00690726 0.999976i $$-0.497801\pi$$
0.00690726 + 0.999976i $$0.497801\pi$$
$$618$$ 0 0
$$619$$ 14.3431 0.576500 0.288250 0.957555i $$-0.406927\pi$$
0.288250 + 0.957555i $$0.406927\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ −3.02944 −0.121372
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 6.82843 0.272701
$$628$$ 0 0
$$629$$ −13.6569 −0.544534
$$630$$ 0 0
$$631$$ −45.6569 −1.81757 −0.908785 0.417264i $$-0.862989\pi$$
−0.908785 + 0.417264i $$0.862989\pi$$
$$632$$ 0 0
$$633$$ −1.17157 −0.0465658
$$634$$ 0 0
$$635$$ −14.4853 −0.574831
$$636$$ 0 0
$$637$$ 35.7157 1.41511
$$638$$ 0 0
$$639$$ −2.34315 −0.0926934
$$640$$ 0 0
$$641$$ 6.97056 0.275321 0.137660 0.990479i $$-0.456042\pi$$
0.137660 + 0.990479i $$0.456042\pi$$
$$642$$ 0 0
$$643$$ −37.9411 −1.49625 −0.748126 0.663557i $$-0.769046\pi$$
−0.748126 + 0.663557i $$0.769046\pi$$
$$644$$ 0 0
$$645$$ −8.82843 −0.347619
$$646$$ 0 0
$$647$$ −4.68629 −0.184237 −0.0921186 0.995748i $$-0.529364\pi$$
−0.0921186 + 0.995748i $$0.529364\pi$$
$$648$$ 0 0
$$649$$ 4.00000 0.157014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6.97056 0.272779 0.136390 0.990655i $$-0.456450\pi$$
0.136390 + 0.990655i $$0.456450\pi$$
$$654$$ 0 0
$$655$$ 3.31371 0.129477
$$656$$ 0 0
$$657$$ 11.3137 0.441390
$$658$$ 0 0
$$659$$ 15.3137 0.596537 0.298269 0.954482i $$-0.403591\pi$$
0.298269 + 0.954482i $$0.403591\pi$$
$$660$$ 0 0
$$661$$ 9.31371 0.362261 0.181131 0.983459i $$-0.442024\pi$$
0.181131 + 0.983459i $$0.442024\pi$$
$$662$$ 0 0
$$663$$ 6.62742 0.257388
$$664$$ 0 0
$$665$$ 5.65685 0.219363
$$666$$ 0 0
$$667$$ −19.3137 −0.747830
$$668$$ 0 0
$$669$$ 6.34315 0.245240
$$670$$ 0 0
$$671$$ −11.6569 −0.450008
$$672$$ 0 0
$$673$$ 18.3431 0.707076 0.353538 0.935420i $$-0.384978\pi$$
0.353538 + 0.935420i $$0.384978\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 29.4558 1.13208 0.566040 0.824378i $$-0.308475\pi$$
0.566040 + 0.824378i $$0.308475\pi$$
$$678$$ 0 0
$$679$$ −9.65685 −0.370596
$$680$$ 0 0
$$681$$ 14.0000 0.536481
$$682$$ 0 0
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 0 0
$$685$$ 13.3137 0.508691
$$686$$ 0 0
$$687$$ −2.00000 −0.0763048
$$688$$ 0 0
$$689$$ −52.6863 −2.00719
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 0 0
$$693$$ −0.828427 −0.0314693
$$694$$ 0 0
$$695$$ 0.485281 0.0184078
$$696$$ 0 0
$$697$$ −5.65685 −0.214269
$$698$$ 0 0
$$699$$ −18.8284 −0.712157
$$700$$ 0 0
$$701$$ 36.1421 1.36507 0.682535 0.730853i $$-0.260878\pi$$
0.682535 + 0.730853i $$0.260878\pi$$
$$702$$ 0 0
$$703$$ 79.5980 3.00209
$$704$$ 0 0
$$705$$ −4.00000 −0.150649
$$706$$ 0 0
$$707$$ 0.686292 0.0258106
$$708$$ 0 0
$$709$$ 6.68629 0.251109 0.125554 0.992087i $$-0.459929\pi$$
0.125554 + 0.992087i $$0.459929\pi$$
$$710$$ 0 0
$$711$$ −8.48528 −0.318223
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 5.65685 0.211554
$$716$$ 0 0
$$717$$ −17.6569 −0.659407
$$718$$ 0 0
$$719$$ 47.5980 1.77511 0.887553 0.460706i $$-0.152404\pi$$
0.887553 + 0.460706i $$0.152404\pi$$
$$720$$ 0 0
$$721$$ −2.74517 −0.102235
$$722$$ 0 0
$$723$$ −12.3431 −0.459047
$$724$$ 0 0
$$725$$ −4.82843 −0.179323
$$726$$ 0 0
$$727$$ −33.9411 −1.25881 −0.629403 0.777079i $$-0.716701\pi$$
−0.629403 + 0.777079i $$0.716701\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −10.3431 −0.382555
$$732$$ 0 0
$$733$$ −6.34315 −0.234289 −0.117145 0.993115i $$-0.537374\pi$$
−0.117145 + 0.993115i $$0.537374\pi$$
$$734$$ 0 0
$$735$$ 6.31371 0.232885
$$736$$ 0 0
$$737$$ 5.65685 0.208373
$$738$$ 0 0
$$739$$ 15.1127 0.555930 0.277965 0.960591i $$-0.410340\pi$$
0.277965 + 0.960591i $$0.410340\pi$$
$$740$$ 0 0
$$741$$ −38.6274 −1.41901
$$742$$ 0 0
$$743$$ −36.3431 −1.33330 −0.666650 0.745371i $$-0.732273\pi$$
−0.666650 + 0.745371i $$0.732273\pi$$
$$744$$ 0 0
$$745$$ −1.51472 −0.0554950
$$746$$ 0 0
$$747$$ 10.0000 0.365881
$$748$$ 0 0
$$749$$ 14.3431 0.524087
$$750$$ 0 0
$$751$$ −20.2843 −0.740184 −0.370092 0.928995i $$-0.620674\pi$$
−0.370092 + 0.928995i $$0.620674\pi$$
$$752$$ 0 0
$$753$$ −20.9706 −0.764210
$$754$$ 0 0
$$755$$ 16.4853 0.599961
$$756$$ 0 0
$$757$$ −36.6274 −1.33125 −0.665623 0.746288i $$-0.731834\pi$$
−0.665623 + 0.746288i $$0.731834\pi$$
$$758$$ 0 0
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ −28.8284 −1.04503 −0.522515 0.852630i $$-0.675006\pi$$
−0.522515 + 0.852630i $$0.675006\pi$$
$$762$$ 0 0
$$763$$ −14.3431 −0.519257
$$764$$ 0 0
$$765$$ 1.17157 0.0423583
$$766$$ 0 0
$$767$$ −22.6274 −0.817029
$$768$$ 0 0
$$769$$ 10.6863 0.385358 0.192679 0.981262i $$-0.438282\pi$$
0.192679 + 0.981262i $$0.438282\pi$$
$$770$$ 0 0
$$771$$ −16.3431 −0.588584
$$772$$ 0 0
$$773$$ 3.65685 0.131528 0.0657640 0.997835i $$-0.479052\pi$$
0.0657640 + 0.997835i $$0.479052\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −9.65685 −0.346438
$$778$$ 0 0
$$779$$ 32.9706 1.18129
$$780$$ 0 0
$$781$$ −2.34315 −0.0838443
$$782$$ 0 0
$$783$$ −4.82843 −0.172554
$$784$$ 0 0
$$785$$ −18.0000 −0.642448
$$786$$ 0 0
$$787$$ 20.1421 0.717990 0.358995 0.933340i $$-0.383120\pi$$
0.358995 + 0.933340i $$0.383120\pi$$
$$788$$ 0 0
$$789$$ −18.0000 −0.640817
$$790$$ 0 0
$$791$$ 15.7157 0.558787
$$792$$ 0 0
$$793$$ 65.9411 2.34164
$$794$$ 0 0
$$795$$ −9.31371 −0.330323
$$796$$ 0 0
$$797$$ 34.9706 1.23872 0.619360 0.785107i $$-0.287392\pi$$
0.619360 + 0.785107i $$0.287392\pi$$
$$798$$ 0 0
$$799$$ −4.68629 −0.165789
$$800$$ 0 0
$$801$$ 3.65685 0.129209
$$802$$ 0 0
$$803$$ 11.3137 0.399252
$$804$$ 0 0
$$805$$ 3.31371 0.116793
$$806$$ 0 0
$$807$$ 20.6274 0.726119
$$808$$ 0 0
$$809$$ 28.4264 0.999419 0.499710 0.866193i $$-0.333440\pi$$
0.499710 + 0.866193i $$0.333440\pi$$
$$810$$ 0 0
$$811$$ −0.485281 −0.0170405 −0.00852027 0.999964i $$-0.502712\pi$$
−0.00852027 + 0.999964i $$0.502712\pi$$
$$812$$ 0 0
$$813$$ 11.7990 0.413809
$$814$$ 0 0
$$815$$ −7.31371 −0.256188
$$816$$ 0 0
$$817$$ 60.2843 2.10908
$$818$$ 0 0
$$819$$ 4.68629 0.163752
$$820$$ 0 0
$$821$$ −12.8284 −0.447715 −0.223858 0.974622i $$-0.571865\pi$$
−0.223858 + 0.974622i $$0.571865\pi$$
$$822$$ 0 0
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 0 0
$$825$$ 1.00000 0.0348155
$$826$$ 0 0
$$827$$ −41.3137 −1.43662 −0.718309 0.695724i $$-0.755084\pi$$
−0.718309 + 0.695724i $$0.755084\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ −2.34315 −0.0812828
$$832$$ 0 0
$$833$$ 7.39697 0.256290
$$834$$ 0 0
$$835$$ −13.3137 −0.460740
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 22.6274 0.781185 0.390593 0.920564i $$-0.372270\pi$$
0.390593 + 0.920564i $$0.372270\pi$$
$$840$$ 0 0
$$841$$ −5.68629 −0.196079
$$842$$ 0 0
$$843$$ −11.1716 −0.384769
$$844$$ 0 0
$$845$$ −19.0000 −0.653620
$$846$$ 0 0
$$847$$ −0.828427 −0.0284651
$$848$$ 0 0
$$849$$ 8.82843 0.302991
$$850$$ 0 0
$$851$$ 46.6274 1.59837
$$852$$ 0 0
$$853$$ −8.68629 −0.297413 −0.148706 0.988881i $$-0.547511\pi$$
−0.148706 + 0.988881i $$0.547511\pi$$
$$854$$ 0 0
$$855$$ −6.82843 −0.233527
$$856$$ 0 0
$$857$$ −28.4853 −0.973039 −0.486519 0.873670i $$-0.661734\pi$$
−0.486519 + 0.873670i $$0.661734\pi$$
$$858$$ 0 0
$$859$$ 52.9706 1.80733 0.903666 0.428238i $$-0.140865\pi$$
0.903666 + 0.428238i $$0.140865\pi$$
$$860$$ 0 0
$$861$$ −4.00000 −0.136320
$$862$$ 0 0
$$863$$ 20.6863 0.704170 0.352085 0.935968i $$-0.385473\pi$$
0.352085 + 0.935968i $$0.385473\pi$$
$$864$$ 0 0
$$865$$ −2.82843 −0.0961694
$$866$$ 0 0
$$867$$ −15.6274 −0.530735
$$868$$ 0 0
$$869$$ −8.48528 −0.287843
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ 0 0
$$873$$ 11.6569 0.394525
$$874$$ 0 0
$$875$$ 0.828427 0.0280059
$$876$$ 0 0
$$877$$ 2.62742 0.0887216 0.0443608 0.999016i $$-0.485875\pi$$
0.0443608 + 0.999016i $$0.485875\pi$$
$$878$$ 0 0
$$879$$ −6.82843 −0.230317
$$880$$ 0 0
$$881$$ −46.9706 −1.58248 −0.791239 0.611507i $$-0.790564\pi$$
−0.791239 + 0.611507i $$0.790564\pi$$
$$882$$ 0 0
$$883$$ 5.37258 0.180802 0.0904009 0.995905i $$-0.471185\pi$$
0.0904009 + 0.995905i $$0.471185\pi$$
$$884$$ 0 0
$$885$$ −4.00000 −0.134459
$$886$$ 0 0
$$887$$ 15.6569 0.525706 0.262853 0.964836i $$-0.415337\pi$$
0.262853 + 0.964836i $$0.415337\pi$$
$$888$$ 0 0
$$889$$ −12.0000 −0.402467
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 0 0
$$893$$ 27.3137 0.914018
$$894$$ 0 0
$$895$$ −17.6569 −0.590204
$$896$$ 0 0
$$897$$ −22.6274 −0.755507
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −10.9117 −0.363521
$$902$$ 0 0
$$903$$ −7.31371 −0.243385
$$904$$ 0 0
$$905$$ 14.0000 0.465376
$$906$$ 0 0
$$907$$ −40.9706 −1.36041 −0.680203 0.733024i $$-0.738108\pi$$
−0.680203 + 0.733024i $$0.738108\pi$$
$$908$$ 0 0
$$909$$ −0.828427 −0.0274772
$$910$$ 0 0
$$911$$ 48.9706 1.62247 0.811234 0.584722i $$-0.198797\pi$$
0.811234 + 0.584722i $$0.198797\pi$$
$$912$$ 0 0
$$913$$ 10.0000 0.330952
$$914$$ 0 0
$$915$$ 11.6569 0.385364
$$916$$ 0 0
$$917$$ 2.74517 0.0906534
$$918$$ 0 0
$$919$$ −11.5147 −0.379836 −0.189918 0.981800i $$-0.560822\pi$$
−0.189918 + 0.981800i $$0.560822\pi$$
$$920$$ 0 0
$$921$$ 3.17157 0.104507
$$922$$ 0 0
$$923$$ 13.2548 0.436288
$$924$$ 0 0
$$925$$ 11.6569 0.383275
$$926$$ 0 0
$$927$$ 3.31371 0.108836
$$928$$ 0 0
$$929$$ −45.5980 −1.49602 −0.748011 0.663687i $$-0.768991\pi$$
−0.748011 + 0.663687i $$0.768991\pi$$
$$930$$ 0 0
$$931$$ −43.1127 −1.41296
$$932$$ 0 0
$$933$$ 3.31371 0.108486
$$934$$ 0 0
$$935$$ 1.17157 0.0383145
$$936$$ 0 0
$$937$$ 11.0294 0.360316 0.180158 0.983638i $$-0.442339\pi$$
0.180158 + 0.983638i $$0.442339\pi$$
$$938$$ 0 0
$$939$$ 15.6569 0.510942
$$940$$ 0 0
$$941$$ −34.7696 −1.13346 −0.566728 0.823905i $$-0.691791\pi$$
−0.566728 + 0.823905i $$0.691791\pi$$
$$942$$ 0 0
$$943$$ 19.3137 0.628941
$$944$$ 0 0
$$945$$ 0.828427 0.0269487
$$946$$ 0 0
$$947$$ −6.62742 −0.215362 −0.107681 0.994185i $$-0.534343\pi$$
−0.107681 + 0.994185i $$0.534343\pi$$
$$948$$ 0 0
$$949$$ −64.0000 −2.07753
$$950$$ 0 0
$$951$$ −26.2843 −0.852326
$$952$$ 0 0
$$953$$ −11.7990 −0.382207 −0.191103 0.981570i $$-0.561207\pi$$
−0.191103 + 0.981570i $$0.561207\pi$$
$$954$$ 0 0
$$955$$ 5.65685 0.183052
$$956$$ 0 0
$$957$$ −4.82843 −0.156081
$$958$$ 0 0
$$959$$ 11.0294 0.356159
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −17.3137 −0.557926
$$964$$ 0 0
$$965$$ −13.6569 −0.439630
$$966$$ 0 0
$$967$$ −11.4558 −0.368395 −0.184198 0.982889i $$-0.558969\pi$$
−0.184198 + 0.982889i $$0.558969\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ 34.6274 1.11125 0.555623 0.831434i $$-0.312480\pi$$
0.555623 + 0.831434i $$0.312480\pi$$
$$972$$ 0 0
$$973$$ 0.402020 0.0128882
$$974$$ 0 0
$$975$$ −5.65685 −0.181164
$$976$$ 0 0
$$977$$ 2.68629 0.0859421 0.0429710 0.999076i $$-0.486318\pi$$
0.0429710 + 0.999076i $$0.486318\pi$$
$$978$$ 0 0
$$979$$ 3.65685 0.116874
$$980$$ 0 0
$$981$$ 17.3137 0.552784
$$982$$ 0 0
$$983$$ 30.6274 0.976863 0.488431 0.872602i $$-0.337569\pi$$
0.488431 + 0.872602i $$0.337569\pi$$
$$984$$ 0 0
$$985$$ 8.48528 0.270364
$$986$$ 0 0
$$987$$ −3.31371 −0.105477
$$988$$ 0 0
$$989$$ 35.3137 1.12291
$$990$$ 0 0
$$991$$ 30.6274 0.972912 0.486456 0.873705i $$-0.338289\pi$$
0.486456 + 0.873705i $$0.338289\pi$$
$$992$$ 0 0
$$993$$ −6.34315 −0.201294
$$994$$ 0 0
$$995$$ −21.6569 −0.686568
$$996$$ 0 0
$$997$$ −39.3137 −1.24508 −0.622539 0.782589i $$-0.713899\pi$$
−0.622539 + 0.782589i $$0.713899\pi$$
$$998$$ 0 0
$$999$$ 11.6569 0.368807
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2640.2.a.bb.1.1 2
3.2 odd 2 7920.2.a.cg.1.1 2
4.3 odd 2 165.2.a.a.1.1 2
12.11 even 2 495.2.a.d.1.2 2
20.3 even 4 825.2.c.e.199.4 4
20.7 even 4 825.2.c.e.199.1 4
20.19 odd 2 825.2.a.g.1.2 2
28.27 even 2 8085.2.a.ba.1.1 2
44.43 even 2 1815.2.a.k.1.2 2
60.23 odd 4 2475.2.c.m.199.1 4
60.47 odd 4 2475.2.c.m.199.4 4
60.59 even 2 2475.2.a.m.1.1 2
132.131 odd 2 5445.2.a.m.1.1 2
220.219 even 2 9075.2.a.v.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.1 2 4.3 odd 2
495.2.a.d.1.2 2 12.11 even 2
825.2.a.g.1.2 2 20.19 odd 2
825.2.c.e.199.1 4 20.7 even 4
825.2.c.e.199.4 4 20.3 even 4
1815.2.a.k.1.2 2 44.43 even 2
2475.2.a.m.1.1 2 60.59 even 2
2475.2.c.m.199.1 4 60.23 odd 4
2475.2.c.m.199.4 4 60.47 odd 4
2640.2.a.bb.1.1 2 1.1 even 1 trivial
5445.2.a.m.1.1 2 132.131 odd 2
7920.2.a.cg.1.1 2 3.2 odd 2
8085.2.a.ba.1.1 2 28.27 even 2
9075.2.a.v.1.1 2 220.219 even 2